# 2.3 | The Derivative of a Function

For any function $$f$$, we define the derivative function, $$f’$$, by
$$f'(x) = \begin{array}{c} \text{Rate of change} \\ \text{ of } f \text{ at } a \end{array} = \lim_{h \rightarrow 0} \frac{f(x+h) -f(x)}{h}.$$
If $$f'(x) > 0$$ on an interval, then $$f$$ is increasing on that interval. If $$f'(x) < 0$$ on an interval, then $$f$$ is decreasing on that interval.
If $$f(x) = k$$, then $$f'(x) = 0$$.
If $$f(x) = mx + b,$$ then $$f'(x) = \text{ Slope } = m.$$
If $$f(x) = x^n,$$ then $$f'(x) = n x^{n-1}.$$